Temperature at the tip of a fin

1. The problem statement, all variables and given/known data

3. The attempt at a solution
I initially found the temperature tip without accounting for the corrected length. But the solution uses the corrected length for finding the temperature which doesn’t make sense to me. Don’t we want the temperature at the tip L, if we find the temperature at the corrected length then that isn’t the ACTUAL temperature, it is an approximation we use to find efficiency I thought?

Here is my solution along with the relevant equations:

Here is the textbook solution:


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